“How quickly does radiation burn?”

Radiation burns are a function of dose. So higher doses, like Radiation therapy, are more likely to cause injury. Lower doses, like from complex X-ray procedures are less likely to injure the skin. A skin injury can be seen during the first four weeks. And sometimes as far out as three months. If you have something you think maybe a burn, show it to your doctor.

This is a complex question beyond the scope of an answer here. For basic procedural Xray's or fluoroscopy there is a threshold dose of about 2000 mGy or 2Gy skin dose that would cause a rash much like a sunburn. This would show up about 3-4 weeks after the exposure. A dose of 5Gy to the skin may result in a 2nd degree skin radiation burn that would also take about 3-4 weeks to show up. If you are instead talking about nuclear waste or then it would depend on the time, dose, and exposure which can vary greatly by factors of 1-1000.